# Equation of a Line

## 1 Equation of a Line 3.4

By the end of this section, you should be able to solve the following problems.

1. Find the equation of the line with the given slope and passing through

the given point.

2. Rewrite the equation in slope-intercept form.

3x − 5y = 4

3. Find the equation of the line passing through the given points in standard

form.

(−2,−3), (1, 2)

4. Find the equation of the line passing through the given points, in
standard

form.

(−2,−3) and (1, 2)

## 2 Concepts

There are three forms of the equation of a line. The most general form is

called standard form. The standard form of an equation of a line is given

below.

Ax + By = C

In the above equation, the coefficient of x can be positive, or zero.
Whenever

the coefficient of x is negative, multiply both sides of the equation by -1

to make it positive. The coefficient of y can be positive, negative, or zero. A

line will exist so long as A and B are not both zero at the same time. The

second form of the equation of a line is called the point-slope form of the

equation of a line, and it is given below:

y − y_{1} = m(x − x_{1})

In the equation above, m stands for the slope and (x_{1},
y_{1}) designate a

point. The third equation is called the slope-intercept form of the equation

and it is given below.

y = mx + b

In the above equation m stands for the slope and b is the
y-intercept. In

our first example, we build the equation of a line using a point and a slope.

**2.1 Example**

Find the equation of a line with the given slope and
passing through a given

point.

Since we are given a point and a slope, we substitute into
the point-slope

form of the equation of a line.

Substituting (-1,4) for x_{1} and y_{1}:

Simplifying:

Multiplying both side of the equation by 5 to eliminate denominators.

In our next example we rewrite an equation in slope-intercept form.

**2.2 Example**

Rewrite the following equation in slope intercept form.

Dividing both sides by 4 we have:

Distributing the denominator we have:

In our last example, we construct a line using two points.

**2.3 Example**

Find the equation of a line that passes through the
following points, and

write it in standard form.

(−1, 5) and (2, 1)

First we find the slope of the line.

Substituting into the point-slope equation we have.

Simplifying:

Multiplying both sides by 3:

Clearing fractions gives:

## 3 Facts

1. Than standard form of the equation of a line is:

Ax + By = C

2. The point-slope form of the equation of a line is:

y − y_{1} = m(x − x_{1})

3. The slope-intercept from of the equation of a line is:

y = mx + b

## 4 Exercises

1. Find the equation of the line with the given slope and
passing through

the given point.

2. Rewrite the equation in slope-intercept form.

3x − 5y = 4

3. Find the equation of the line passing through the given
points, in standard

form.

(−1, 2) and (2, 3)

4. Find the equation of the line passing through the given
points, in standard

form.

(−2,−3) and (1, 2)

## 5 Solutions

1. Find the equation of the line with the given slope and
passing through

the given point.

2. Rewrite the equation in slope-intercept form.

3. Find the equation of the line passing through the given
points, in standard

form.

(−1, 2) and (2, 3)

4. Find the equation of the line passing through the given
points, in standard

form.

(−2,−3) and (1, 2)

−5x + 3y = 1

(−1)(−5x + 3y) = (−1)1

5x − 3y = −1